Average Error: 27.1 → 16.6
Time: 19.3s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -4.963679822741098923711711123984814712826 \cdot 10^{90}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;y \le 1.495453019463623917434530780565366229939 \cdot 10^{124}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, x, \mathsf{fma}\left(a, t, \left(\left(a + z\right) - b\right) \cdot y\right)\right)}{\left(y + t\right) + x}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array}\]
\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -4.963679822741098923711711123984814712826 \cdot 10^{90}:\\
\;\;\;\;\left(a + z\right) - b\\

\mathbf{elif}\;y \le 1.495453019463623917434530780565366229939 \cdot 10^{124}:\\
\;\;\;\;\frac{\mathsf{fma}\left(z, x, \mathsf{fma}\left(a, t, \left(\left(a + z\right) - b\right) \cdot y\right)\right)}{\left(y + t\right) + x}\\

\mathbf{else}:\\
\;\;\;\;\left(a + z\right) - b\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r42914549 = x;
        double r42914550 = y;
        double r42914551 = r42914549 + r42914550;
        double r42914552 = z;
        double r42914553 = r42914551 * r42914552;
        double r42914554 = t;
        double r42914555 = r42914554 + r42914550;
        double r42914556 = a;
        double r42914557 = r42914555 * r42914556;
        double r42914558 = r42914553 + r42914557;
        double r42914559 = b;
        double r42914560 = r42914550 * r42914559;
        double r42914561 = r42914558 - r42914560;
        double r42914562 = r42914549 + r42914554;
        double r42914563 = r42914562 + r42914550;
        double r42914564 = r42914561 / r42914563;
        return r42914564;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r42914565 = y;
        double r42914566 = -4.963679822741099e+90;
        bool r42914567 = r42914565 <= r42914566;
        double r42914568 = a;
        double r42914569 = z;
        double r42914570 = r42914568 + r42914569;
        double r42914571 = b;
        double r42914572 = r42914570 - r42914571;
        double r42914573 = 1.495453019463624e+124;
        bool r42914574 = r42914565 <= r42914573;
        double r42914575 = x;
        double r42914576 = t;
        double r42914577 = r42914572 * r42914565;
        double r42914578 = fma(r42914568, r42914576, r42914577);
        double r42914579 = fma(r42914569, r42914575, r42914578);
        double r42914580 = r42914565 + r42914576;
        double r42914581 = r42914580 + r42914575;
        double r42914582 = r42914579 / r42914581;
        double r42914583 = r42914574 ? r42914582 : r42914572;
        double r42914584 = r42914567 ? r42914572 : r42914583;
        return r42914584;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original27.1
Target11.9
Herbie16.6
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.581311708415056427521064305370896655752 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.228596430831560895857110658734089400289 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -4.963679822741099e+90 or 1.495453019463624e+124 < y

    1. Initial program 46.6

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]

    2. Simplified46.6

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, x, \mathsf{fma}\left(a, t, \left(\left(z + a\right) - b\right) \cdot y\right)\right)}{x + \left(y + t\right)}}\]

    3. Taylor expanded around 0 13.7

      \[\leadsto \color{blue}{\left(a + z\right) - b}\]

    if -4.963679822741099e+90 < y < 1.495453019463624e+124

    1. Initial program 18.0

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]

    2. Simplified18.0

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, x, \mathsf{fma}\left(a, t, \left(\left(z + a\right) - b\right) \cdot y\right)\right)}{x + \left(y + t\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.963679822741098923711711123984814712826 \cdot 10^{90}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;y \le 1.495453019463623917434530780565366229939 \cdot 10^{124}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, x, \mathsf{fma}\left(a, t, \left(\left(a + z\right) - b\right) \cdot y\right)\right)}{\left(y + t\right) + x}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x y z t a b)
  :name "AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1"

  :herbie-target
  (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) -3.5813117084150564e+153) (- (+ z a) b) (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) 1.2285964308315609e+82) (/ 1.0 (/ (+ (+ x t) y) (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)))) (- (+ z a) b)))

  (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))